Exponential Distribution Model

In probability theory and statistics, the exponential distribution (a.k.a. negative exponential distribution) is the probability distribution that describes the time between events in a Poisson process, i.e. a process in which events occur continuously and independently at a constant average rate. It is a particular case of the gamma distribution. It is the continuous analogue of the geometric distribution, and it has the key property of being memoryless. In addition to being used for the analysis of Poisson processes, it is found in various other contexts.

The exponential distribution is not the same as the class of exponential families of distributions, which is a large class of probability distributions that includes the exponential distribution as one of its members, but also includes the normal distribution, binomial distribution, gamma distribution, Poisson, and many others.

The exponential distribution is used to model the time between the occurrence of events in an interval of time, or the distance between events in space. The exponential distribution may be useful to model events such as

The time between goals scored in a World Cup soccer match

The duration of a phone call to a help center

The time between meteors greater than 1 meter diameter striking earth

The time between successive failures of a machine

The time from diagnosis until death in patients with metastatic cancer

The distance between successive breaks in a pipeline

The exponential distribution is an appropriate model if the following conditions are true.

X is the time (or distance) between events, with X > 0.

The occurrence of one event does not affect the probability that a second event will occur. That is, events occur independently.

The rate at which events occur is constant. The rate cannot be higher in some intervals and lower in other intervals.

Two events cannot occur at exactly the same instant.

If these conditions are true, then X is an exponential random variable, and the distribution of X is an exponential distribution. If these conditions are not true, then the exponential distribution is not appropriate. Alternative distributions such as the Weibull or gamma may give a better fit to the data, or a semi-parametric model, such as the Cox proportional-hazards model, may be required for statistical analysis.

The graph of an exponential distribution starts on the y-axis at a positive value (called lambda, λ) and decreases to the right.

The exponential distribution is specified by the single parameter lambda (λ). Lambda is the event rate, and may have different names in other applications:

event rate

rate parameter

arrival rate

death rate

failure rate

transition rate

Lambda is the number of events per unit time. The graph of the exponential distribution, called the probability density function (PDF), shows the distribution of time (or distance) between events. The PDF is specified in terms of lambda (events per unit time) and x (time).

where

λ is the rate parameter

x > 0

e is the number 2.71828 …, the base of the natural logs

The density f(x) is 0 for x less than or equal to 0.

Graphs of exponential distribution PDFs with various lambda
The figure shows the PDF for exponential distributions with lambda = 1, 2, 3, and 0.5. The line for each distribution meets the y-axis at lambda. Notice in the figure that when lambda (the event rate) is large, the time between events is small. In particular, the mean time between events is given by 1/lambda. For lambda = 3, the mean time between events is 1/lambda = 1/3. For lambda = 0.5, the mean time between events is 1/lambda = 1/0.5 = 2.

Olkin gives the following example of data modeled using an exponential distribution. The time between successive failures of the air-conditioning system of a particular jet airplane were recorded:

he mean time between failures is 59.6. The median time between failures is 22. Because the exponential distribution is skewed right, the median is less than the mean. The figure shows a histogram of the time between failures and a fitted exponential density curve with lambda = 1/(mean time to failure) = 1/59.6 = 0.0168.

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